Pub Date : 1992-02-28DOI: 10.1070/IM1992V039N01ABEH002229
S. G. Tankeev
Let be a Kuga-Satake abelian variety defined over a number field . Assuming a certain arithmetic condition on the canonical field associated to , we prove the Mumford-Tate conjecture concerning the Lie algebra of the image of the -adic representation in the one-dimensional cohomology of .
{"title":"Kuga-Satake Abelian Varieties and l-ADIC Representations","authors":"S. G. Tankeev","doi":"10.1070/IM1992V039N01ABEH002229","DOIUrl":"https://doi.org/10.1070/IM1992V039N01ABEH002229","url":null,"abstract":"Let be a Kuga-Satake abelian variety defined over a number field . Assuming a certain arithmetic condition on the canonical field associated to , we prove the Mumford-Tate conjecture concerning the Lie algebra of the image of the -adic representation in the one-dimensional cohomology of .","PeriodicalId":159459,"journal":{"name":"Mathematics of The Ussr-izvestiya","volume":"03 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122483952","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1992-02-28DOI: 10.1070/IM1992V038N01ABEH002192
B. Khabibullin
This paper presents a technique for constructing functions that are subharmonic in the complex plane, agree with a given subharmonic function u on a system S of rays with vertex at the origin, and are harmonic outside S. For a wide class of systems S, this technique permits one to obtain criteria for the complete regularity of growth of entire functions f on S in terms of the balayage of the distribution of zeros of f.
{"title":"BALAYAGE ON A SYSTEM OF RAYS AND ENTIRE FUNCTIONS OF COMPLETELY REGULAR GROWTH","authors":"B. Khabibullin","doi":"10.1070/IM1992V038N01ABEH002192","DOIUrl":"https://doi.org/10.1070/IM1992V038N01ABEH002192","url":null,"abstract":"This paper presents a technique for constructing functions that are subharmonic in the complex plane, agree with a given subharmonic function u on a system S of rays with vertex at the origin, and are harmonic outside S. For a wide class of systems S, this technique permits one to obtain criteria for the complete regularity of growth of entire functions f on S in terms of the balayage of the distribution of zeros of f.","PeriodicalId":159459,"journal":{"name":"Mathematics of The Ussr-izvestiya","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116827938","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1992-02-28DOI: 10.1070/IM1992V039N01ABEH002224
A. Fomenko
Some new objects, bordisms of integrable systems, are found and studied. The classes of rigidly bordant systems form a nontrivial abelian group, which makes it possible to construct new integrable systems on the basis of previously known ones. Among the generators of this bordism group are known physical integrable systems, as, for example, the Lagrange system (from the dynamics of a heavy rigid body) and others. Moreover, a new topological invariant of systems with many degrees of freedom is also constructed. It turns out that two integrable systems are topologically equivalent if and only if their invariants coincide. In particular, it follows from this that the set of topological classes of integrable systems is discrete. The invariant can be effectively calculated for concrete integrable systems arising in physics and mechanics.
{"title":"A BORDISM THEORY FOR INTEGRABLE NONDEGENERATE HAMILTONIAN SYSTEMS WITH TWO DEGREES OF FREEDOM. A NEW TOPOLOGICAL INVARIANT OF HIGHER-DIMENSIONAL INTEGRABLE SYSTEMS","authors":"A. Fomenko","doi":"10.1070/IM1992V039N01ABEH002224","DOIUrl":"https://doi.org/10.1070/IM1992V039N01ABEH002224","url":null,"abstract":"Some new objects, bordisms of integrable systems, are found and studied. The classes of rigidly bordant systems form a nontrivial abelian group, which makes it possible to construct new integrable systems on the basis of previously known ones. Among the generators of this bordism group are known physical integrable systems, as, for example, the Lagrange system (from the dynamics of a heavy rigid body) and others. Moreover, a new topological invariant of systems with many degrees of freedom is also constructed. It turns out that two integrable systems are topologically equivalent if and only if their invariants coincide. In particular, it follows from this that the set of topological classes of integrable systems is discrete. The invariant can be effectively calculated for concrete integrable systems arising in physics and mechanics.","PeriodicalId":159459,"journal":{"name":"Mathematics of The Ussr-izvestiya","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130200539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1992-02-28DOI: 10.1070/IM1992V038N01ABEH002188
P. Dudnikov, S. Samborskii
For an arbitrary endomorphism of the free semimodule over an Abelian semiring with operations and it is shown under the assumption that is idempotent (and under certain other restrictions on ) that there exists a nontrivial "spectrum", i.e., there exist a and a nontrivial subsemimodule such that for any . The same result is also obtained for endomorphism analogues of integral operators (in the sense of the theory of idempotent integration). In terms of this spectrum investigations are made of the asymptotic behavior of endomorphisms under iteration and of convergence of the "Neumann series" appearing in the solution of the equations . The simplest examples are connected with the semiring and arise, for example, in dynamic programming problems.
{"title":"ENDOMORPHISMS OF SEMIMODULES OVER SEMIRINGS WITH AN IDEMPOTENT OPERATION","authors":"P. Dudnikov, S. Samborskii","doi":"10.1070/IM1992V038N01ABEH002188","DOIUrl":"https://doi.org/10.1070/IM1992V038N01ABEH002188","url":null,"abstract":"For an arbitrary endomorphism of the free semimodule over an Abelian semiring with operations and it is shown under the assumption that is idempotent (and under certain other restrictions on ) that there exists a nontrivial \"spectrum\", i.e., there exist a and a nontrivial subsemimodule such that for any . The same result is also obtained for endomorphism analogues of integral operators (in the sense of the theory of idempotent integration). In terms of this spectrum investigations are made of the asymptotic behavior of endomorphisms under iteration and of convergence of the \"Neumann series\" appearing in the solution of the equations . The simplest examples are connected with the semiring and arise, for example, in dynamic programming problems.","PeriodicalId":159459,"journal":{"name":"Mathematics of The Ussr-izvestiya","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123511154","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1992-02-28DOI: 10.1070/IM1992V038N01ABEH002191
A. Petukhov
This paper studies the dependence of the properties of a function on the degree of its approximation by piecewise monotone functions (in particular, by rational functions and polynomials) in the uniform and Hausdorff metrics, i.e., inverse theorems in the theory of approximation. Of interest are properties related to the Hausdorff dimension and measure of the set of points of discontinuity and to the graph of a function.
{"title":"ON THE DEPENDENCE OF PROPERTIES OF THE GRAPH OF A FUNCTION ON THE DEGREE OF VARIOUS APPROXIMATIONS","authors":"A. Petukhov","doi":"10.1070/IM1992V038N01ABEH002191","DOIUrl":"https://doi.org/10.1070/IM1992V038N01ABEH002191","url":null,"abstract":"This paper studies the dependence of the properties of a function on the degree of its approximation by piecewise monotone functions (in particular, by rational functions and polynomials) in the uniform and Hausdorff metrics, i.e., inverse theorems in the theory of approximation. Of interest are properties related to the Hausdorff dimension and measure of the set of points of discontinuity and to the graph of a function.","PeriodicalId":159459,"journal":{"name":"Mathematics of The Ussr-izvestiya","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125215387","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1992-02-28DOI: 10.1070/IM1992V038N01ABEH002193
S. Strunkov
We obtain a congruence type arithmetic relation on the set of all triples (G,H,P), where G is a finite group, H is a subgroup, and P is a representation of G by permutations. This relation becomes Fermat's Little Theorem in the case when G=Zp, H=1, and P is the regular representation of G.
{"title":"ON A GENERALIZATION OF FERMAT'S LITTLE THEOREM","authors":"S. Strunkov","doi":"10.1070/IM1992V038N01ABEH002193","DOIUrl":"https://doi.org/10.1070/IM1992V038N01ABEH002193","url":null,"abstract":"We obtain a congruence type arithmetic relation on the set of all triples (G,H,P), where G is a finite group, H is a subgroup, and P is a representation of G by permutations. This relation becomes Fermat's Little Theorem in the case when G=Zp, H=1, and P is the regular representation of G.","PeriodicalId":159459,"journal":{"name":"Mathematics of The Ussr-izvestiya","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124455631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1992-02-28DOI: 10.1070/IM1992V039N01ABEH002228
V. Kolyvagin, D. Y. Logachëv
Kolyvagin's method for the proof of the finiteness of is extended to abelian varieties with real multiplication, of -rank 0, defined over totally real fields, if they are factors of the Jacobians of Shimura curves. The finiteness of for such a variety is proved, starting from the conditions that a Heegner point on it is not a torsion point.
{"title":"FINITENESS OF OVER TOTALLY REAL FIELDS","authors":"V. Kolyvagin, D. Y. Logachëv","doi":"10.1070/IM1992V039N01ABEH002228","DOIUrl":"https://doi.org/10.1070/IM1992V039N01ABEH002228","url":null,"abstract":"Kolyvagin's method for the proof of the finiteness of is extended to abelian varieties with real multiplication, of -rank 0, defined over totally real fields, if they are factors of the Jacobians of Shimura curves. The finiteness of for such a variety is proved, starting from the conditions that a Heegner point on it is not a torsion point.","PeriodicalId":159459,"journal":{"name":"Mathematics of The Ussr-izvestiya","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131440461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1992-02-28DOI: 10.1070/IM1992V038N01ABEH002195
A. S. Merkur’ev
This paper gives a criterion for the splittability of a central simple finitedimensional division algebra under the extension of the ground field to the universal splitting field of a certain quadratic form. The criterion is stated in terms of the Clifford algebra of the quadratic form. As a corollary, it is shown that the u-invariant of a field can take any even values.
{"title":"SIMPLE ALGEBRAS AND QUADRATIC FORMS","authors":"A. S. Merkur’ev","doi":"10.1070/IM1992V038N01ABEH002195","DOIUrl":"https://doi.org/10.1070/IM1992V038N01ABEH002195","url":null,"abstract":"This paper gives a criterion for the splittability of a central simple finitedimensional division algebra under the extension of the ground field to the universal splitting field of a certain quadratic form. The criterion is stated in terms of the Clifford algebra of the quadratic form. As a corollary, it is shown that the u-invariant of a field can take any even values.","PeriodicalId":159459,"journal":{"name":"Mathematics of The Ussr-izvestiya","volume":"30 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120850198","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1992-02-28DOI: 10.1070/IM1992V038N01ABEH002189
I. Kuzin
Under suitable conditions countable solvability of the problem in , , 0$ SRC=http://ej.iop.org/images/0025-5726/38/1/A05/tex_im_2189_img4.gif/>, , where is a ball of radius , is proved.
{"title":"EXISTENCE OF A COUNTABLE SET OF PERIODIC, SPHERICALLY SYMMETRIC SOLUTIONS OF A NONLINEAR WAVE EQUATION","authors":"I. Kuzin","doi":"10.1070/IM1992V038N01ABEH002189","DOIUrl":"https://doi.org/10.1070/IM1992V038N01ABEH002189","url":null,"abstract":"Under suitable conditions countable solvability of the problem in , , 0$ SRC=http://ej.iop.org/images/0025-5726/38/1/A05/tex_im_2189_img4.gif/>, , where is a ball of radius , is proved.","PeriodicalId":159459,"journal":{"name":"Mathematics of The Ussr-izvestiya","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115119565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1992-02-28DOI: 10.1070/IM1992V039N01ABEH002222
A. Veretennikov
The logarithmic asymptotic behavior is established for large deviations of the "slow" motion in the averaging principle for stochastic equations in a finite-dimensional Euclidean space.
建立了有限维欧几里德空间随机方程平均原理中“慢”运动大偏差的对数渐近性。
{"title":"ON LARGE DEVIATIONS IN THE AVERAGING PRINCIPLE FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS. II","authors":"A. Veretennikov","doi":"10.1070/IM1992V039N01ABEH002222","DOIUrl":"https://doi.org/10.1070/IM1992V039N01ABEH002222","url":null,"abstract":"The logarithmic asymptotic behavior is established for large deviations of the \"slow\" motion in the averaging principle for stochastic equations in a finite-dimensional Euclidean space.","PeriodicalId":159459,"journal":{"name":"Mathematics of The Ussr-izvestiya","volume":"104 5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131298863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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